On Darboux-Treibich-Verdier potentials

TL;DR

This paper demonstrates that a specific four-parameter family of elliptic functions encompasses the most general class of meromorphic potentials with infinitely many finite-gap solutions, extending Darboux, Treibich, and Verdier's work.

Contribution

It identifies the four-parameter family as the most general meromorphic family with infinitely many finite-gap potentials, unifying previous results.

Findings

The family includes infinitely many finite-gap potentials.

It generalizes earlier known potentials by Darboux, Treibich, and Verdier.

The family is characterized by specific elliptic functions.

Abstract

It is shown that the four-parameter family of elliptic functions $$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$$ introduced by Darboux and rediscovered a hundred years later by Treibich and Verdier, is the most general meromorphic family containing infinitely many finite-gap potentials.